Not to be taken seriously. The first thing I noticed was a schoolboy blunder in celestial mechanics: you don’t add velocities in the way that he did, you add energies, that’s to say the squares of the velocities. If an asteroid with an asymptotic approach speed of 5.6 km/s is going to hit Earth its impact speed (prior to air resistance) would be 12.5 km/s, not 17 km/s. We can therefore safely ignore the rest of his story. — Noted Astronomer
Fortunately, when it comes to orbital calculations and intuitive dynamic visualizations of the solar system the Tusk is more than its author. Can any readers analyze and verify the claims made by the Scute at Tall Bloke?
I will acknowledge that when NASA said that the Chebarkul meteor was not following along the same path as 2012DA14 they were right but only in a highly technical sense. Any fragment passing within a radius of a few thousand miles of the trajectory would not have impacted the other side of the Earth. But when considering the possibility of meteor showers, you have to think in terms of millions of miles, even for asteroidal showers such as the Geminids and the Quarantids, because the Earth takes days to travel through them. I think NASA was clutching at straws. You have to look at the bigger picture and besides, the proposition as put forward in this comment isn’t played out on a vast scale in solar system terms. — Andrew Copper at talk bloke
The Chelyabinsk Meteor and a possible link with 2012DA14
I think the idea of the Russian meteor being related to 2012DA14 should be resurrected. I say resurrected because the idea was so roundly slapped down by NASA within hours of the impact and never discussed again. Most of the information below was gleaned from NASA’s own JPL Horizons ephemeris for 2012DA14.
Let me begin by addressing a few myths that seemed to sew it up regarding the lack of any link between the two
Firstly, the direction of approach was not on the night side of the earth but on the day side (2012DA14 flipped under and up round the back only in the last 5 hours) and the radiant was not, as variously described, “the South Pole” or -81 degrees (implied by the above as being -81 to the night side), but at -69 degrees on the sunward side.
Secondly, the radiant had a right ascension of almost exactly 00 hours ,that is, 30 degrees east of the sun (which was at 21 hours 54 min of RA on the day) in the equatorial plane. The Russian (Chebarkul) meteor came in at 13 degrees east of the sun in local horizontal coordinates.
Thirdly, the incoming trajectory of the meteor was not north-south but on an azimuth of 99 degrees i.e. 9 degrees south of east. Since it was sunrise this meant that the meteor came from a direction close to the sun (13 degrees east of it), in other words, coming in over a great circle running down the globe to the south, although a better approximation would be south east, This was possible because the Earth’s axis was tilted back by 12.5 on that date, making a late sunrise for Chebarkul, so watching the sunrise on a somewhat tighter, northern latitude line meant looking along a straight line that soon scribed south eastwards in lower latitudes (rather than curving round the 55 degree North line).
Fourthly, 2012DA14 was not going “too slow” for a related fragment to arrive at 17km/second: its radiant, relative velocity to the Earth before being accelerated was 12600mph. That is 5.6 km/ sec. If you add to that the freefall velocity of 11.2 km/sec (the corollary of escape velocity) you get 16.8 km/sec. Add to that the eastward rotation of the earth at 55degrees north at an Azimuth of 9 degrees south of east (0.2 km/sec) you arrive at precisely 17km/sec. This is the same calculation that Zuluaga and Ferrin (and now, NASA) must have done in reverse for their version of the reconstruction of the trajectory: I calculated the radial speed of their hypothesised orbits at the Earth’s position (r value/ radius from sun=1AU) on the day of impact (but without the Earth’s gravitational influence added) and ended up with 34.8 and 35.2 km/sec for the 2 posited orbits. That amounts to 5 and 5.4 km/sec relative to the Earth, respectively. Adding the freefall velocity and the eastward rotation you get 16.4 and 16.8km/sec. The difference between these posited orbits and the posited 2012DA14 fragment is that they invoke the head-on trajectory solution with little or no curvature as they are pulled into the gravity well. If it’s a bulls-eye hit the curvature is zero. The Zuluaga and Ferrin video shows the meteor coming in from about 3 degrees above the solar plane. The NASA video now shows the same.
I believe there are multiple solutions between head on (3 degree inclination) and an 11.6 degree inclination (11.6 deg being the solar plane analogue to the -69 degree radiant from a geocentric view). These various solutions involve increasing degrees of curvature as the meteor is pulled into the Earth’s gravitational well but curvature in freefall doesn’t make the final velocity any slower. By the way, none of these high curvature scenarios would involve capture in orbit- it’s either impact or escape along a hyperbolic path back out of the well. And high curvature means a 50 degree ‘q angle’ (half way through the turn). It’s the q angle that determines how far round to the north the meteor can curve and still impact rather than miss and escape.
The last myth is that at 55 degrees latitude Cherbarkul is too far North for any fragments to hit. This is not true because from a -69 degree radiant, the ‘equator line’, or tangent line, that 2012DA14 could see from below was shunted upward (from the real equator) on the sunward side by 21 degrees. Due to the vagaries of the trajectory of any purported fragment (see below), it would still require around 55 degrees of hyperbolic turn from that raised equator line onwards, although that describes a track all the way to Chebarkul- atmospheric entry would begin at 50 degrees round. For an idea of the sort of curvature needed for the fragment, I plotted a curve from first principles (subtracting the freefall component of velocity from the baseline, straight velocity of 17km/sec) and came up with a q angle of 50 degrees. For some perspective, the Apollo missions came in on a hyperbolic orbit at 11.2 km/sec and still got 69 degrees round the back of the globe, regardless of rotation and a sedate 1080 mile reentry. Even 2012DA14 curved 30 degrees or more for a q angle of 15 deg and that was under the influence of one tenth the gravity. 50 degrees is probably an upper limit but that is exactly what is required for a 2012DA14 fragment riding along on the radiant angle to turn in hyperbolically and hit Chebarkul at a low trajectory.
One other point, though not classifiable as a myth is that 2012DA14 is being characterised by NASA as a CO or CV type chondrite (carbonaceous with calcium and aluminium inclusions) based on spectral observations whereas they say the Chebarkul meteor is a stony chondrite because the few pieces found so far “are reported to be silicate rich”. This, they say, rules out any link. This may be true but the whole tenor of their delivery is one of running scared and has to be looked at in the light of the following quotes, all within a minute on one video:
[re 2012DA14] “there was no danger of a collisions, NASA assured people”
“… In a one in a million chance that still has NASA scientists shaking their heads”.
“These are rare events and it’s incredible to see them happen on the same day.”
As things stand as of 5th March 2013 , I feel that the evidence presented here is more convincing than some spectral measurements not chiming with a few reported silicate bearing fragments.
The radiant is on the right at 00hours 30M. The purported fragment didn’t have to be on this line , just slightly displaced and parallel to it a few hours back up range at say 80-100,000 miles.
So we now have a completely different, sunward radiant of -69 degrees of declination. In fact, for a fragment travelling 50,000 miles sunward from the track line of 2012DA14 and 16 hours ahead, it would probably come in from nearer to a -66 to -67 degree trajectory before being really hiked round on its hyperbolic orbit (its close encounter-to-impact trajectory). 2012DA14 itself turned in by 2-3 degrees from its side of the track in the last day before starting its 30 degree hyperbolic turn. A -66 or -67 incoming trajectory for the hypothetical fragment means the raised ‘equator line’ was up to 24 degrees higher on the sunward side, reducing the required curvature even further, possibly to 47 degrees before atmospheric entry. Indeed, that sunward fragment track, displaced as it is seemingly arbitrarily, at 50,000 miles over and parallel to 2012DA14 actually allows for 10-12,000 miles or 2 degrees of inward curvature from 03:00 on 14th February, 24 hours up range. This is before being really hiked in from a -66 to -67 degree point at about 22:00 UTC on February 14th, some 5 hours from impact.
You can download the ephemeris for 2012DA14 from JPL horizons website (click ‘web interface’ and enter ‘observer’ ‘geocentric’ and dates from 1st Feb 2013 to 28th Feb 2013 at hourly intervals). I suggest the entire month so as to give a better feel of what’s going on. It’s easy to scroll up and down quickly. It doesn’t show speeds but I got the figure of 12600 mph from a news item and checked it against the orbital speed and inclination of 2012DA14 for the vertical component, then derived the horizontal geocentric component from that. Those vectors do pretty well add up to a 12600mph, 69 degree slope until one or two days out (300k to 600k miles). The JPL video has 2012DA14 at about 13,500 mph at 4 hours out. I apologise for the lack of links, they are currently playing havoc with my formatting. I might do another separate comment with some links.
I have described the trajectory of the hypothetical fragment. Now I need to describe where to look for it in a forensic sense- using astrodynamics software to rerun different scenarios with the meteor exiting from a narrow window around the proposed path. If anyone here has astrodynamics software, please feel free to join in and prove it one way or the other. I have no software so feedback would be welcome.
The best place to look for the fragment is emerging from a window bounded by a geocentric declination of between -59 and -65 degrees and a right ascension of between 22H and 2H 30 minutes. The declination angles of the window aperture are less than the radiant and trajectory angle because the fragment is now cutting up through the angle lines, past the radiant, trying to get level with the earth in the same way that 2012DA14 cut the other way through the angles to come up the back: the radiant becomes irrelevant at this point and that’s why it was totally misleading to talk about a -81 degree radiant.
Also, because the fragment is cutting up through the declination angles, its own trajectory angle cannot be described with declination angles any more- except for a bulls eye geocentric hit. It has to be aiming over the top of the Earth to get a chance of being pulled in hyperbolically over the ‘new’ equator line for a hit. This means that whatever declination angle is chosen for the instantaneous position of the emergence of the fragment through the window, its actual trajectory angle would need to be greater by 3 to 6 degrees or so in order to aim it away from the geocentre to two or three thousand miles out from the equator line, that is, two or three thousand miles from the disc it sees above it and with the same right ascension as the RA of the fragment emerging at the window. This angle cannot be geocentric because it has to look as if it will miss the Earth. Looked at another way, it is a line running parallel to a declination angle that starts 3 to 6 degrees in from the fragment emergence point, measured radially inward. This jiggling of inputs for the fragments own inclination is nevertheless bounded by the upper limit of 69 deg, the true radiant angle. There is also potentially a small amount of leftward (solar plane y axis) component as they emerge round towards the 2H 30 mark because these are trying to pass by the side of the Earth but get caught. I think these, around the 0 to 1H mark (0 to 30 degrees of right ascension) are the best candidates.
That arrival window describes a curved slit sitting somewhere roughly below where the southern tip of New Zealand was at the time of impact. The fragment would be emerging at a trajectory angle of between -65 and -69 degrees of declination (that is, its own path angle as opposed to the box perimeter angles). This needs to be set at 10 hours 20 mins (clock time) up range (about 150k to 170k miles?) so that the fragment would be coming through it at 17:00 UTC on February 14th 2013. Some time not long after that the trajectory angle would start to curve in noticeably on its hyperbolic path.
I extended the window round to 2H 30M because of the angle discrepancy between the radiant and the final trajectory. This was up to 17 degrees when playing with equatorial-to-horizontal coords for Chebarkul (but one-way calcs made this inexact) and only 8.5 degrees using old fashioned cotton stuck to a globe.
Incidentally, using the globe method resulted in a view, looking downrange along the trajectory, identical to the several Youtube videos of 2012DA14 when freeze-framed at 3:20 UTC on 15th February. It was the view from DA14 as the Chebarkul meteor hit- you can see the beginning of the meteor’s ground track over Taiwan before it disappears over the ‘equator line’ at 22 deg north. You can see that its hyperbolic trajectory extends round and down past and almost parallel to you to your right, in other words, the same trajectory but displaced 50k miles to the right. This ‘almost parallel’ track would correspond to the sunward track of the fragment not being quite sunward but with a slight right ascension from that track of a few thousand miles so that the fragment came from further round towards 2H 30 but up and in on the correct line. This allows for the 8.5 degree disparity.
It should be said here that 2012DA14 seems to have corkscrewed itself by between 5 and 12.5 degrees depending on where you make the cutoff between true radiant and local approach. It is apparent in the ephemeris (you can’t see its subtlety in any video or diagram). It may seem to defy physics (not orbiting on a great circle) but I would ascribe it to the slowing down 2012DA14 as its orbit goes from inside track to outside track and its relative prograde motion with respect to the Earth went nearly to zero. It was almost stationary in the prograde vector, sitting at -86 declination and ready to be plucked up along the most convenient longitude line. Because there was some forward motion still, it did get plucked up pretty well opposite but not 180 degrees- the entire pass was 4 or 5 degrees inside one hemisphere, resulting in a 5 to 12.5 degree twist. If this happened to the proposed Chebarkul fragment, it would have skewed in exactly the necessary way to bring it round to this new ‘wrong’ 8.5 degree-off trajectory and heading over Taiwan, China, and into Chebarkul. This is why the search box extends round to 2H 30M. It’s because the fragment oversteps ever so slightly before skewing. For that reason, once you get round to 00H to 2H, the trajectory emerging from the box will have a sideways component to the left (positive RA) ie not radially inward. These candidates will be skewing 8-10 degrees anticlockwise as you look down on them as they rise and will probably do so over the last 80,000 miles. This is how their local ‘radiant’ is skewed round. That’s if DA14 is anything to go by.
The fragment would be travelling at about 13,000 mph through the window but it would be best to plug into the software ephemeris details for 2012DA14 for the speeds from far out so as not to start at an arbitrarily high speed. However, beware of piggy-backing on 2012DA14 data to extrapolate hypothesised fragment data. I have seen, among other things, a rather amusing graph that relied on like-for-like parallel trajectories with earth skimming fragments refusing to bend round under the influence of 10 x the acceleration 2012DA14 was experiencing. That is their ‘proof’ that the fragments can’t make it north of the equator. That was on a respected astronomy blog.
If all goes to plan, you should see all manner of near misses, flying over Asia and Russia, a few direct hits on the Southern Hemisphere in a wide band from New Zealand northwards and a few Northern Hemisphere hits, one of them right on Chebarkul. You’ll have to play with it though, maybe venture a little way outside the window if needs be. The worst that can happen is some very interesting near misses- but I really do think there will be hits. This need for playing around is reflected in the fact that some solutions could imply the need to add 5 or 10,000 miles to the track line displacement further uprange (back down the slope) for a possible 60,000mile or so displacement. You can’t use the 2012DA14 trajectory, displaced and pasted to the other track- it’s bending the wrong way from 24 hours out. Even a mirror image wouldn’t be faithful due to its greater radial displacement.
THE POSITION OF THE FRAGMENT IN ORBIT- THE BIGGER PICTURE:
I will acknowledge that when NASA said that the Chebarkul meteor was not following along the same path as 2012DA14 they were right but only in a highly technical sense. Any fragment passing within a radius of a few thousand miles of the trajectory would not have impacted the other side of the Earth. But when considering the possibility meteor showers, you have to think in terms of millions of miles, even for asteroidal showers such as the Geminids and the Quarantids, because the Earth takes days to travel through them. I think NASA was clutching at straws. You have to look at the bigger picture and besides, the proposition as put forward in this comment isn’t played out on a vast scale in solar system terms.
I propose a fragment riding just 200,000 miles above 2012DA14 (north with respect to the solar plane). Just stating that baldly might understandably invite querying as to why they should be related. However, visualising it scaled down, it would be the equivalent of two tiny pieces of rock on an orbit 54 metres round, mirroring each other’s every move in speed, inclination and eccentricity, all the while staying exactly 2 centimetres apart, one directly above the other. I would consider those two pieces as related, one broken off from the other.
When the 65,000 mph prograde element of the Earth’s orbit is removed we get the geocentric element of relative movement between 2012DA14 (along with its hypothetical fragment) and Earth. Gone is the gentle 10 degree inclination with respect to the Earth’s orbit, with the asteroid climbing gently up a slope and sedately past the night side. It’s turned into a precipitous 69 degree climb, skewing round to vertical as 2012DA14 was apparently dragged up from under the South Pole and slung shot vertically above. That does serve a purpose for geocentric calculations and visualisations but it is as well to remember that it helps to plug mentally into the elongated, gentle slope version from time to time so as to get a good feel for what is really happening to the Asteroid and the purported fragment as they pass Earth.
Once the frame of reference snaps to geocentric you see the fragment rising up ahead of 2012DA14 and slightly to one side and they start to look a little disjointed. But when you snap back to solar system view with them both sailing along, rising up alongside Earth, you see that one is directly above the other: the 50,000 mile displacement is really just a 200,000 mile vertical displacement which, when looking down the -69 degree radiant makes them appear to be 50,000 miles apart. That said, there may be a fraction of further displacement to the outside of the orbit track too to allow for the skewing effect. They only ride on different tracks because of their vertical displacement and those two tracks went either side of the Earth. One was too close and the fragment hit (hypothetically). There’s something telling about that vertical nature of the relationship: the proposed fragment is following the exact same track but directly above. If it had been shifted a few thousand miles long ago, perhaps by a small collision, then previous close encounters, passing beneath the Earth would have widened that gap quite dramatically: 0.01mm/sec^2 differentials in gravitational acceleration add up to 4m/sec over a four day encounter within a million miles. That’s tens of thousands of miles per year. 200,000 miles isn’t as far as it seems.
I doubt if its possible but old sky scans might show up the culprit: at 12 million miles it would be 15 minutes of arc displaced from 2012DA14 when looking straight down the 69 degree track and would be offset at around the ’9 o’clock’ mark. For the 2012 pass at 6 million miles it would have been around one degree offset above at about the 12 o’clock mark.